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Shaped & Holey Boards . KNIGHT'S TOUR NOTES 3 Shaped & Holey Boards by G. P. Jelliss 2019 1 . KNIGHT'S TOUR NOTES ======================================================================== Title Page Illustrations: Biaxial tours of 12 and 20 cell boards, the first of each being by Euler (1759). The asymmetric 16-cell tour covers a quarter of the standard 8×8 chessboard. The large diagram is of a 320-cell tour by the Rajah of Mysore. The original, in a form of Indian Script, is shown on page 20 of Das Schach und seine asiatischen Verwandten (2008) by Maria Schetelich. The central 8×8 tour is the Rajah's magic tour, numbered with 1 at f2 and 64 at d1. At the centres of the 7-move stars in the outer border 20 cells of the 18×18 board are omitted. The tour as a whole is closed, cell 320 being a knight move from cell 1. Rotary tours of a 24-cell board and of 76-cell boards formed by removing 24 cells from a 10×10 board, externally or internally. --------------------------------------------------------------------------------------------------------------------------------- Contents Octonary 3. Number of cells 8 to 80 in steps of 8 Birotary 7. Number of cells 16 to 196 in steps of 4 Biaxial 24. Number of cells 10 to 52 in steps of 2 Rotary (Eulerian and Bergholtian) 32. Number of cells 13 to 248 Axial (Sulian and Murraian) 53. Number of cells 14 to 124 Unary (Asymmetric) 64. Number of cells 13 to 160 79. © George Peter.Jelliss 2019 http://www.mayhematics.com/ Knight's Tour Notes, Volume 3, Shaped & Holey Boards. If cited in other works please give due acknowledgment of the source as for a normal book. 2 . KNIGHT'S TOUR NOTES ======================================================================== Octonary ======================================================================== A pattern within a square area that appears unaltered by any of the transformations which leave the square unaltered has octonary symmetry. The diagonals and medians devide the pattern into eight triangles all of the same pattern. Such symmetry is also termed ‘square’ or ‘perfect square’ symmetry. For the cells to be considered a ‘board’ they must be properly connected edge to edge. Cells can be omitted in fours on the diagonals and medians, or in eights when off-axis. Tours with octonary symmetry thus all use a multiple of 8 cells (4 diagonal, 4 medial, each other position in sets of 8). A knight tour with octonary symmetry on any even-sided board, such as the 8×8 chessboard, is impossible. In fact tours with octonary symmetry are not possible within any even-sided area, even with holes. Octonary tours all require boards within a containing square of odd side and omit the centre cell. Every octonary tour consists of eight equal paths, each path having one end on a median cell and the other end on a diagonal cell. 8 cells: The smallest board with a closed knight tour is of course the 8-cell centreless 3×3 board. In our coding system the centre cell is 0, the mid-edge cells are 1, and the corner cells are 2, so the moves are described by the formula: 1-2. The earliest manuscripts associated with al-Adli and as-Suli (c.900) include the puzzle, often attributed to the later writer Guarini (1512), of the four knights. White and Black knights placed in the corners of the 3×3 board are to be interchanged by knight moves without having two on a cell at the same time. The knights go in procession round the star-shaped closed tour of the eight edge cells (taking 4 not 2 moves each). NN n n 16 cells: The next case is a 16-cell tour within a 5×5 area, leaving a cross-shaped hole, tour formula 2-4-3. I call this ‘the bound cross’. 24 cells: The shortest octonary tours within the 7×7 area are two with 24 cells, both having the formula 1-7-4-2, where the 7-4 move can be taken in two different ways. We can write it 7:4 or 7;4 according as the move crosses a median or diagonal. 9 8 7 6 7 8 9 8 5 4 3 4 5 8 7 4 2 1 2 4 7 6 3 1 0 1 3 6 7 4 2 1 2 4 7 8 5 4 3 4 5 8 9 8 7 6 7 8 9 The inset diagram above shows how the cells are coded from the centre outwards. The 7-4 or 4-7 moves are the only cases where the coding of a move is ambiguous. 32 cells: There are 6 other octonary tours within the 7×7 area, on two board shapes, covering 32 cells. Their formulas are 3-4-7-8-2, 3-8-7-4-2, 3-8-7-4-9 with two of each according as the move 7-4 is across diagonal or median. 3 . KNIGHT'S TOUR NOTES There are another 8 octonary tours of 32 cells within the 9×9 area using three board shapes. The formulas including the 7-4 link are 3-11-7-4-2, 10-4-7-11-5, 3-4-7-11-5 occurring in pairs. The other two formulae are 10-8-7-11-5 and 10-8-7-11-9 on two board shapes, whose cells become ever more tenuously connected. 4 . KNIGHT'S TOUR NOTES 40 cells: These five examples using 40 cells are all within a 9×9 area. One pair has the formula 6-12-4-7-11-5 (with 4-7 taken two ways). The other three are 6-12-8-7-11-5, 6-12-8-7-11-9 (differing only at cells 5 and 9) and 6-12-13-7-11-9. 48 cells: Here are three octonary tours of 48 cells within a 9×9 area. A pair with formula 10-8-12-4-7-11-5, and another with formula 10-8-12-13-7-11-9. 56 cells: A pair of octonary tours of 56 cells in an 11×11 area. Formula: 1-7-4-13-12-18-8-2. 5 . KNIGHT'S TOUR NOTES 64 cells: Octonary tour of 64 cells within an 11×11 area. Formula: 3-4-12-8-18-11-7-17-9. 72 cells: Octonary tour of 72 cells within the 11×11 area. Formula 15-12-13-16-8-18-11-7-17-9. 80 cells: A pair of octonary tours of 80 cells, the maximum possible within the 11×11 area. Formula: 3-8-18-11-16-13-7-4-12-19-9. These are the largest examples of this symmetry that I have constructed. 6 . KNIGHT'S TOUR NOTES ======================================================================== Birotary ======================================================================== Birotary symmetry, also known as ‘oblique quaternary symmetry’, is altered by reflection, but not by 90° rotation. This is also possible in tours on Square Boards of sides 6, 10, 14, 18, etc. (see 5). 16 cells: Two birotary tours on a board formed of four 2×2 boards, surrounding a single-cell hole. 20 cells: Four birotary tours (Jelliss 2013). Two on one board. One with four holes. 24 cells: Four birotary tours on a 5×5 with centre hole and with corner cells moved one step diagonally. The four corner cells cannot however be added at the middle of each side, to give a board with octonary symmetry, since this fails to maintain the correct balance of colours when chequered. 28 cells: Within a 6×6 area there are 10 birotary tours covering 28 cells, using four shapes of board, one with four single holes in a skew pattern. 7 . KNIGHT'S TOUR NOTES 32 cells: Three boards of assorted shapes, with holes, showing 90 degree rotation. 36 cells: Birotary symmetry is possible on the 6×6 board. Here are some alternative 36-cell boards on which birotary tours are also possible. These boards can be regarded as 4×4 with a pentomino added on each edge or corner. There are other cases the reader might like to investigate. Quaternary symmetry is also possible on 36-cell holey boards. This board shape is used by the Rajah of Mysore (#45 in Harikrishna 1871) but his tour (not shown here) is asymmetric. We add two with four single cell holes, and another pentomino example without holes. 8 . KNIGHT'S TOUR NOTES 40 cells: Two assorted holey boards with birotary tours. 44 cells: Three octonary board-shapes can be formed by adding two cells to each side of the 6×6, making 44 cells, and each of these can be toured in birotary symmetry. In the three further tours on the first board (Jelliss 24 May 2013) there are only four acute angles (as shown by the dots). Here is an oblique quaternary board with four oblique quaternary tours (Jelliss 2013). 9 . KNIGHT'S TOUR NOTES 48 cells: According to W. H. Cozens (1940) there are 64 tours with 90° rotary symmetry on the 7×7 centreless board. The first five quaternary tours here are formed from a set of closed quarter-tours of the board by ‘simple-linking’ (a further 15 asymmetric tours can be formed from the same set by this process). The first two are by Kraitchik 1927 and the other three are my own work. These are followed by two other examples from Kraitchik 1927. The last is from Bergholt memorandum 24 Feb 1916. Kraitchik Kraitchin Jelliss Jelliss Jelliss Kraitchik Kraitchk Bergholt 52 cells: Here are three 90° rotatory tours on boards formed from the 6 ×6 with four cells added to each edge in various formations (Jelliss 2013). The fourth example, below, with a T-tetromino added to each edge, is by S.Vatriquant (L'Echiquier 1928).
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